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In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, the closure of a subset of points in a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
consists of all
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
s in together with all
limit points In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...
of . The closure of may equivalently be defined as the
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
of and its
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
, and also as the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of all
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
s containing . Intuitively, the closure can be thought of as all the points that are either in or "near" . A point which is in the closure of is a
point of closure In mathematics, an adherent point (also closure point or point of closure or contact point) Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15. of a subset A of a topological space X, is a point x in X such that every neighbourhood of x (or equivalen ...
of . The notion of closure is in many ways dual to the notion of interior.


Definitions


Point of closure

For S as a subset of a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, x is a point of closure of S if every
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
centered at x contains a point of S (this point can be x itself). This definition generalizes to any subset S of a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
X. Fully expressed, for X as a metric space with metric d, x is a point of closure of S if for every r > 0 there exists some s \in S such that the distance d(x, s) < r (x = s is allowed). Another way to express this is to say that x is a point of closure of S if the distance d(x, S) := \inf_ d(x, s) = 0 where \inf is the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
. This definition generalizes to
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s by replacing "open ball" or "ball" with "
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
". Let S be a subset of a topological space X. Then x is a or of S if every neighbourhood of x contains a point of S (again, x = s for s \in S is allowed). Note that this definition does not depend upon whether neighbourhoods are required to be open.


Limit point

The definition of a point of closure is closely related to the definition of a
limit point of a set In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
. The difference between the two definitions is subtle but important – namely, in the definition of a limit point x of a set S, every neighbourhood of x must contain a point of S . (Each neighbourhood of x has x but it also must have a point of S that is different from x.) A limit point of S has more strict condition than a point of closure of S in the definitions. The set of all limit points of a set S is called the . A limit point of a set is also called ''cluster point'' or ''accumulation point'' of the set. Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure which is not a limit point is an
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...
. In other words, a point x is an isolated point of S if it is an element of S and there is a neighbourhood of x which contains no other points of S than x itself. For a given set S and point x, x is a point of closure of S if and only if x is an element of S or x is a limit point of S (or both).


Closure of a set

The of a subset S of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
(X, \tau), denoted by \operatorname_ S or possibly by \operatorname_X S (if \tau is understood), where if both X and \tau are clear from context then it may also be denoted by \operatorname S, \overline, or S ^ (Moreover, \operatorname is sometimes capitalized to \operatorname.) can be defined using any of the following equivalent definitions:
  1. \operatorname S is the set of all Adherent point, points of closure of S.
  2. \operatorname S is the set S together with all of its limit points.
  3. \operatorname S is the intersection of all
    closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
    s containing S.
  4. \operatorname S is the smallest closed set containing S.
  5. \operatorname S is the union of S and its
    boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
    \partial(S).
  6. \operatorname S is the set of all x \in X for which there exists a
    net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
    (valued) in S that converges to x in (X, \tau).
The closure of a set has the following properties. * \operatorname S is a closed superset of S. * The set S is closed
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
S = \operatorname S. * If S \subseteq T then \operatorname S is a subset of \operatorname T. * If A is a closed set, then A contains S if and only if A contains \operatorname S. Sometimes the second or third property above is taken as the of the topological closure, which still make sense when applied to other types of closures (see below). In a
first-countable space In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
(such as a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
), \operatorname S is the set of all
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of all convergent
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s of points in S. For a general topological space, this statement remains true if one replaces "sequence" by "
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
" or "
filter Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component tha ...
" (as described in the article on
filters in topology Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters, which are special families of subsets of some give ...
). Note that these properties are also satisfied if "closure", "superset", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "subset", "union", "contained in", "largest", and "open". For more on this matter, see
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are dete ...
below.


Examples

Consider a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
in a 3 dimensional space. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-
ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
). It is useful to distinguish between the interior and the surface of the sphere, so we distinguish between the open 3-ball (the interior of the sphere), and the closed 3-ball – the closure of the open 3-ball that is the open 3-ball plus the surface (the surface as the sphere itself). In
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
: * In any space, \varnothing = \operatorname \varnothing. In other words, the closure of the empty set \varnothing is \varnothing itself. * In any space X, X = \operatorname X. Giving \mathbb and \mathbb the standard (metric) topology: * If X is the Euclidean space \mathbb of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, then \operatorname_X ((0, 1)) =
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math>. In other words., the closure of the set (0,1) as a subset of X is ,1/math>. * If X is the Euclidean space \mathbb, then the closure of the set \mathbb of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s is the whole space \mathbb. We say that \mathbb is
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in \mathbb. * If X is the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
\mathbb = \mathbb^2, then \operatorname_X \left( \ \right) = \. * If S is a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
subset of a Euclidean space X, then \operatorname_X S = S. (For a general topological space, this property is equivalent to the T1 axiom.) On the set of real numbers one can put other topologies rather than the standard one. * If X = \mathbb is endowed with the
lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of in ...
, then \operatorname_X ((0, 1)) = , 1). * If one considers on X = \mathbb the discrete topology in which every set is closed (open), then \operatorname_X ((0, 1)) = (0, 1). * If one considers on X = \mathbb the trivial topology in which the only closed (open) sets are the empty set and \mathbb itself, then \operatorname_X ((0, 1)) = \mathbb. These examples show that the closure of a set depends upon the topology of the underlying space. The last two examples are special cases of the following. * In any
discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
, since every set is closed (and also open), every set is equal to its closure. * In any
indiscrete space In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
X, since the only closed sets are the empty set and X itself, we have that the closure of the empty set is the empty set, and for every non-empty subset A of X, \operatorname_X A = X. In other words, every non-empty subset of an indiscrete space is
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
. The closure of a set also depends upon in which space we are taking the closure. For example, if X is the set of rational numbers, with the usual
relative topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced by the Euclidean space \mathbb, and if S = \, then S is both closed and open in \mathbb because neither S nor its complement can contain \sqrt2, which would be the lower bound of S, but cannot be in S because \sqrt2 is irrational. So, S has no well defined closure due to boundary elements not being in \mathbb. However, if we instead define X to be the set of real numbers and define the interval in the same way then the closure of that interval is well defined and would be the set of all greater than \sqrt2.


Closure operator

A on a set X is a mapping of the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of X, \mathcal(X), into itself which satisfies the
Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first forma ...
. Given a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
(X, \tau), the topological closure induces a function \operatorname_X : \wp(X) \to \wp(X) that is defined by sending a subset S \subseteq X to \operatorname_X S, where the notation \overline or S^ may be used instead. Conversely, if \mathbb is a closure operator on a set X, then a topological space is obtained by defining the
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
s as being exactly those subsets S \subseteq X that satisfy \mathbb(S) = S (so complements in X of these subsets form the
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s of the topology). The closure operator \operatorname_X is dual to the interior operator, which is denoted by \operatorname_X, in the sense that :\operatorname_X S = X \setminus \operatorname_X (X \setminus S), and also :\operatorname_X S = X \setminus \operatorname_X (X \setminus S). Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators by replacing sets with their complements in X. In general, the closure operator does not commute with intersections. However, in a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ...
the following result does hold:


Facts about closures

A subset S is closed in X if and only if \operatorname_X S = S. In particular: * The closure of the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
is the empty set; * The closure of X itself is X. * The closure of an
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of sets is always a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of (but need not be equal to) the intersection of the closures of the sets. * In a
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
of
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
ly many sets, the closure of the union and the union of the closures are equal; the union of zero sets is the empty set, and so this statement contains the earlier statement about the closure of the empty set as a special case. * The closure of the union of infinitely many sets need not equal the union of the closures, but it is always a
superset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of the union of the closures. ** Thus, just as the union of two closed sets is closed, so too does closure distribute over binary unions: that is, \operatorname_X (S \cup T) = (\operatorname_X S) \cup (\operatorname_X T). But just as a union of infinitely many closed sets is not necessarily closed, so too does closure not necessarily distribute over infinite unions: that is, \operatorname_X \left(\bigcup_ S_i\right) \neq \bigcup_ \operatorname_X S_i is possible when I is infinite. If S \subseteq T \subseteq X and if T is a subspace of X (meaning that T is endowed with the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
that X induces on it), then \operatorname_T S \subseteq \operatorname_X S and the closure of S computed in T is equal to the intersection of T and the closure of S computed in X: \operatorname_T S ~=~ T \cap \operatorname_X S. Because \operatorname_X S is a closed subset of X, the intersection T \cap \operatorname_X S is a closed subset of T (by definition of the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
), which implies that \operatorname_T S \subseteq T \cap \operatorname_X S (because \operatorname_T S is the closed subset of T containing S). Because \operatorname_T S is a closed subset of T, from the definition of the subspace topology, there must exist some set C \subseteq X such that C is closed in X and \operatorname_T S = T \cap C. Because S \subseteq \operatorname_T S \subseteq C and C is closed in X, the minimality of \operatorname_X S implies that \operatorname_X S \subseteq C. Intersecting both sides with T shows that T \cap \operatorname_X S \subseteq T \cap C = \operatorname_T S. \blacksquare It follows that S \subseteq T is a dense subset of T
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
T is a subset of \operatorname_X S. It is possible for \operatorname_T S = T \cap \operatorname_X S to be a proper subset of \operatorname_X S; for example, take X = \R, S = (0, 1), and T = (0, \infty). If S, T \subseteq X but S is not necessarily a subset of T then only \operatorname_T (S \cap T) ~\subseteq~ T \cap \operatorname_X S is always guaranteed, where this containment could be strict (consider for instance X = \R with the usual topology, T = (-\infty, 0], and S = (0, \infty)From T := (-\infty, 0] and S := (0, \infty) it follows that S \cap T = \varnothing and \operatorname_X S = [0, \infty), which implies \varnothing ~=~ \operatorname_T (S \cap T) ~\neq~ T \cap \operatorname_X S ~=~ \. ), although if T happens to an open subset of X then the equality \operatorname_T (S \cap T) = T \cap \operatorname_X S will hold (no matter the relationship between S and T). Let S, T \subseteq X and assume that T is open in X. Let C := \operatorname_T (T \cap S), which is equal to T \cap \operatorname_X (T \cap S) (because T \cap S \subseteq T \subseteq X). The complement T \setminus C is open in T, where T being open in X now implies that T \setminus C is also open in X. Consequently X \setminus (T \setminus C) = (X \setminus T) \cup C is a closed subset of X where (X \setminus T) \cup C contains S as a subset (because if s \in S is in T then s \in T \cap S \subseteq \operatorname_T (T \cap S) = C), which implies that \operatorname_X S \subseteq (X \setminus T) \cup C. Intersecting both sides with T proves that T \cap \operatorname_X S \subseteq T \cap C = C. The reverse inclusion follows from C \subseteq \operatorname_X (T \cap S) \subseteq \operatorname_X S. \blacksquare Consequently, if \mathcal is any
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
of X and if S \subseteq X is any subset then: \operatorname_X S = \bigcup_ \operatorname_U (U \cap S) because \operatorname_U (S \cap U) = U \cap \operatorname_X S for every U \in \mathcal (where every U \in \mathcal is endowed with the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced on it by X). This equality is particularly useful when X is a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
and the sets in the open cover \mathcal are domains of
coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
s. In words, this result shows that the closure in X of any subset S \subseteq X can be computed "locally" in the sets of any open cover of X and then unioned together. In this way, this result can be viewed as the analogue of the well-known fact that a subset S \subseteq X is closed in X if and only if it is "
locally closed In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if any of the following equivalent conditions are satisfied: * E is the intersection of an open set and a closed set in X. * For each point x\in E ...
in X", meaning that if \mathcal is any
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
of X then S is closed in X if and only if S \cap U is closed in U for every U \in \mathcal.


Functions and closure


Continuity

A function f : X \to Y between topological spaces is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
if and only if the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of every closed subset of the codomain is closed in the domain; explicitly, this means: f^(C) is closed in X whenever C is a closed subset of Y. In terms of the closure operator, f : X \to Y is continuous if and only if for every subset A \subseteq X, f\left(\operatorname_X A\right) ~\subseteq~ \operatorname_Y (f(A)). That is to say, given any element x \in X that belongs to the closure of a subset A \subseteq X, f(x) necessarily belongs to the closure of f(A) in Y. If we declare that a point x is a subset A \subseteq X if x \in \operatorname_X A, then this terminology allows for a
plain English Plain English (or layman's terms) are groups of words that are to be clear and easy to know. It usually avoids the use of rare words and uncommon euphemisms to explain the subject. Plain English wording is intended to be suitable for almost anyone, ...
description of continuity: f is continuous if and only if for every subset A \subseteq X, f maps points that are close to A to points that are close to f(A). Thus continuous functions are exactly those functions that preserve (in the forward direction) the "closeness" relationship between points and sets: a function is continuous if and only if whenever a point is close to a set then the image of that point is close to the image of that set. Similarly, f is continuous at a fixed given point x \in X if and only if whenever x is close to a subset A \subseteq X, then f(x) is close to f(A).


Closed maps

A function f : X \to Y is a (strongly)
closed map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
if and only if whenever C is a closed subset of X then f(C) is a closed subset of Y. In terms of the closure operator, f : X \to Y is a (strongly) closed map if and only if \operatorname_Y f(A) \subseteq f\left(\operatorname_X A\right) for every subset A \subseteq X. Equivalently, f : X \to Y is a (strongly) closed map if and only if \operatorname_Y f(C) \subseteq f(C) for every closed subset C \subseteq X.


Categorical interpretation

One may define the closure operator in terms of universal arrows, as follows. The
powerset In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postu ...
of a set X may be realized as a
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
P in which the objects are subsets and the morphisms are
inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iot ...
s A \to B whenever A is a subset of B. Furthermore, a topology T on X is a subcategory of P with inclusion functor I : T \to P. The set of closed subsets containing a fixed subset A \subseteq X can be identified with the
comma category In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objec ...
(A \downarrow I). This category — also a partial order — then has initial object \operatorname A. Thus there is a universal arrow from A to I, given by the inclusion A \to \operatorname A. Similarly, since every closed set containing X \setminus A corresponds with an open set contained in A we can interpret the category (I \downarrow X \setminus A) as the set of open subsets contained in A, with
terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
\operatorname(A), the interior of A. All properties of the closure can be derived from this definition and a few properties of the above categories. Moreover, this definition makes precise the analogy between the topological closure and other types of closures (for example
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1 ...
), since all are examples of
universal arrow In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...
s.


See also

* * * Closed regular set, a set equal to the closure of their interior * * *


Notes


References


Bibliography

* * * * * * * *


External links

* {{DEFAULTSORT:Closure (Topology) General topology Closure operators